Discreteness of spectrum for the $\overline\partial$-Neumann Laplacian on manifolds of bounded geometry

TL;DR

This paper investigates the spectral properties of the $ar{ abla}$-Neumann Laplacian on Hermitian manifolds with bounded geometry, extending known results from pseudoconvex domains in complex Euclidean spaces.

Contribution

It extends spectral results of the $ar{ abla}$-Neumann Laplacian to Kähler manifolds with bounded geometry, connecting complex analysis and magnetic Schrödinger operator theory.

Findings

Spectral properties similar to pseudoconvex domains hold on certain Kähler manifolds.

Results apply to forms with values in line bundles, linking to magnetic Schrödinger operators.

Conditions for discreteness of spectrum are established under bounded geometry assumptions.

Abstract

For a Hermitian holomorphic vector bundle over a Hermitian manifold, we consider the Dolbeault Laplacian with $\overline\partial$-Neumann boundary conditions, which is a self-adjoint operator on the space of square-integrable differential forms with values in the given holomorphic bundle. We argue that some known results on the spectral properties of this operator on pseudoconvex domains in $\mathbb C^n$ continue to hold on K\"ahler manifolds satisfying certain bounded geometry assumptions. In particular, we will consider the Dolbeault complex for forms with values in a line bundle, where known results from magnetic Schr\"odinger operator theory can be applied.